Continuity in probability


In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.

Definition

Let be a stochastic process in.
The process is continuous in probability when converges in probability to whenever converges to.

Examples and Applications

es are continuous in probability at. Continuity in probability is a sometimes used as one of the defining property for Lévy process. Any process that is continuous in probability and has independent increments has a version that is càdlàg. As a result, some authors immediately define Lévy process as being càdlàg and having independent increments.